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Supervised Learning : Exploring Activation Functions And Backpropagation Gradient Updates for Neural Network Classification

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John Pauline Pineda

March 12, 2024

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• 1. Table of Contents
  • 1.1 Data Background
  • 1.2 Data Description
  • 1.3 Data Quality Assessment
  • 1.4 Data Preprocessing
    • 1.4.1 Data Cleaning
    • 1.4.2 Missing Data Imputation
    • 1.4.3 Outlier Treatment
    • 1.4.4 Collinearity
    • 1.4.5 Shape Transformation
    • 1.4.6 Centering and Scaling
    • 1.4.7 Data Encoding
    • 1.4.8 Preprocessed Data Description
  • 1.5 Data Exploration
    • 1.5.1 Exploratory Data Analysis
    • 1.5.2 Hypothesis Testing
  • 1.6 Neural Network Classification Gradient and Weight Updates
    • 1.6.1 Premodelling Data Description
    • 1.6.2 Sigmoid Activation Function
    • 1.6.3 Rectified Linear Unit Activation Function
    • 1.6.4 Leaky Rectified Linear Unit Activation Function
    • 1.6.5 Exponential Linear Unit Activation Function
    • 1.6.6 Scaled Exponential Linear Unit Activation Function
    • 1.6.7 Randomized Leaky Rectified Linear Unit Activation Function
  • 1.7 Consolidated Findings
• 2. Summary
• 3. References

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1. Table of Contents

This project manually implements the Sigmoid, Rectified Linear Unit, Leaky Rectified Linear Unit, Exponential Linear Unit, Scaled Exponential Linear Unit and Randomized Leaky Rectified Linear Unit activation functions using various helpful packages in Python with fixed values applied for the learning rate and iteration count parameters to optimally update the gradients and weights of an artificial neural network classification model. The gradient, weight, cost function and classification accuracy optimization profiles of the different activation settings were compared. All results were consolidated in a Summary presented at the end of the document.

Artificial Neural Network, in the context of categorical response prediction, consists of interconnected nodes called neurons organized in layers. The model architecture involves an input layer which receives the input data, with each neuron representing a feature or attribute of the data; hidden layers which perform computations on the input data through weighted connections between neurons and apply activation functions to produce outputs; and the output layer which produces the final predictions equal to the number of classes, each representing the probability of the input belonging to a particular class, based on the computations performed in the hidden layers. Neurons within adjacent layers are connected by weighted connections. Each connection has an associated weight that determines the strength of influence one neuron has on another. These weights are adjusted during the training process to enable the network to learn from the input data and make accurate predictions. Activation functions introduce non-linearities into the network, allowing it to learn complex relationships between inputs and outputs. The training process involves presenting input data along with corresponding target outputs to the network and adjusting the weights to minimize the difference between the predicted outputs and the actual targets which is typically performed through optimization algorithms such as gradient descent and backpropagation. The training process iteratively updates the weights until the model's predictions closely match the target outputs.

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

1.1. Data Background

Datasets used for the analysis were separately gathered and consolidated from various sources including:

1. Cancer Rates from World Population Review
2. Social Protection and Labor Indicator from World Bank
3. Education Indicator from World Bank
4. Economy and Growth Indicator from World Bank
5. Environment Indicator from World Bank
6. Climate Change Indicator from World Bank
7. Agricultural and Rural Development Indicator from World Bank
8. Social Development Indicator from World Bank
9. Health Indicator from World Bank
10. Science and Technology Indicator from World Bank
11. Urban Development Indicator from World Bank
12. Human Development Indices from Human Development Reports
13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

The target variable for the study is:

• CANRAT - Dichotomized category based on age-standardized cancer rates, per 100K population (2022)

The predictor variables for the study are:

• GDPPER - GDP per person employed, current US Dollars (2020)
• URBPOP - Urban population, % of total population (2020)
• PATRES - Patent applications by residents, total count (2020)
• RNDGDP - Research and development expenditure, % of GDP (2020)
• POPGRO - Population growth, annual % (2020)
• LIFEXP - Life expectancy at birth, total in years (2020)
• TUBINC - Incidence of tuberculosis, per 100K population (2020)
• DTHCMD - Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
• AGRLND - Agricultural land, % of land area (2020)
• GHGEMI - Total greenhouse gas emissions, kt of CO2 equivalent (2020)
• RELOUT - Renewable electricity output, % of total electricity output (2015)
• METEMI - Methane emissions, kt of CO2 equivalent (2020)
• FORARE - Forest area, % of land area (2020)
• CO2EMI - CO2 emissions, metric tons per capita (2020)
• PM2EXP - PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
• POPDEN - Population density, people per sq. km of land area (2020)
• GDPCAP - GDP per capita, current US Dollars (2020)
• ENRTER - Tertiary school enrollment, % gross (2020)
• HDICAT - Human development index, ordered category (2020)
• EPISCO - Environment performance index , score (2022)

1.2. Data Description

1. The dataset is comprised of:
   • 177 rows (observations)
   • 22 columns (variables)
     • 1/22 metadata (object)
       • COUNTRY
     • 1/22 target (categorical)
       • CANRAT
     • 19/22 predictor (numeric)
       • GDPPER
       • URBPOP
       • PATRES
       • RNDGDP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • RELOUT
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • ENRTER
       • EPISCO
     • 1/22 predictor (categorical)
       • HDICAT

##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
import os
%matplotlib inline

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats

##################################
# Defining file paths
##################################
DATASETS_ORIGINAL_PATH = r"datasets\original"

##################################
# Loading the dataset
# from the DATASETS_ORIGINAL_PATH
##################################
cancer_rate = pd.read_csv(os.path.join("..", DATASETS_ORIGINAL_PATH, "CategoricalCancerRates.csv"))

##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT      object
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()

	COUNTRY	CANRAT	GDPPER	URBPOP	PATRES	RNDGDP	POPGRO	LIFEXP	TUBINC	DTHCMD	...	RELOUT	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	ENRTER	GDPCAP	HDICAT	EPISCO
0	Australia	High	98380.63601	86.241	2368.0	NaN	1.235701	83.200000	7.2	4.941054	...	13.637841	131484.763200	17.421315	14.772658	24.893584	3.335312	110.139221	51722.06900	VH	60.1
1	New Zealand	High	77541.76438	86.699	348.0	NaN	2.204789	82.256098	7.2	4.354730	...	80.081439	32241.937000	37.570126	6.160799	NaN	19.331586	75.734833	41760.59478	VH	56.7
2	Ireland	High	198405.87500	63.653	75.0	1.23244	1.029111	82.556098	5.3	5.684596	...	27.965408	15252.824630	11.351720	6.768228	0.274092	72.367281	74.680313	85420.19086	VH	57.4
3	United States	High	130941.63690	82.664	269586.0	3.42287	0.964348	76.980488	2.3	5.302060	...	13.228593	748241.402900	33.866926	13.032828	3.343170	36.240985	87.567657	63528.63430	VH	51.1
4	Denmark	High	113300.60110	88.116	1261.0	2.96873	0.291641	81.602439	4.1	6.826140	...	65.505925	7778.773921	15.711000	4.691237	56.914456	145.785100	82.664330	60915.42440	VH	77.9

5 rows × 22 columns

##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)

##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:

	count	mean	std	min	25%	50%	75%	max
GDPPER	165.0	45284.424283	3.941794e+04	1718.804896	13545.254510	34024.900890	66778.416050	2.346469e+05
URBPOP	174.0	59.788121	2.280640e+01	13.345000	42.432750	61.701500	79.186500	1.000000e+02
PATRES	108.0	20607.388889	1.340683e+05	1.000000	35.250000	244.500000	1297.750000	1.344817e+06
RNDGDP	74.0	1.197474	1.189956e+00	0.039770	0.256372	0.873660	1.608842	5.354510e+00
POPGRO	174.0	1.127028	1.197718e+00	-2.079337	0.236900	1.179959	2.031154	3.727101e+00
LIFEXP	174.0	71.746113	7.606209e+00	52.777000	65.907500	72.464610	77.523500	8.456000e+01
TUBINC	174.0	105.005862	1.367229e+02	0.770000	12.000000	44.500000	147.750000	5.920000e+02
DTHCMD	170.0	21.260521	1.927333e+01	1.283611	6.078009	12.456279	36.980457	6.520789e+01
AGRLND	174.0	38.793456	2.171551e+01	0.512821	20.130276	40.386649	54.013754	8.084112e+01
GHGEMI	170.0	259582.709895	1.118550e+06	179.725150	12527.487367	41009.275980	116482.578575	1.294287e+07
RELOUT	153.0	39.760036	3.191492e+01	0.000296	10.582691	32.381668	63.011450	1.000000e+02
METEMI	170.0	47876.133575	1.346611e+05	11.596147	3662.884908	11118.976025	32368.909040	1.186285e+06
FORARE	173.0	32.218177	2.312001e+01	0.008078	11.604388	31.509048	49.071780	9.741212e+01
CO2EMI	170.0	3.751097	4.606479e+00	0.032585	0.631924	2.298368	4.823496	3.172684e+01
PM2EXP	167.0	91.940595	2.206003e+01	0.274092	99.627134	100.000000	100.000000	1.000000e+02
POPDEN	174.0	200.886765	6.453834e+02	2.115134	27.454539	77.983133	153.993650	7.918951e+03
ENRTER	116.0	49.994997	2.970619e+01	2.432581	22.107195	53.392460	71.057467	1.433107e+02
GDPCAP	170.0	13992.095610	1.957954e+04	216.827417	1870.503029	5348.192875	17421.116227	1.173705e+05
EPISCO	165.0	42.946667	1.249086e+01	18.900000	33.000000	40.900000	50.500000	7.790000e+01

##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:

	count	unique	top	freq
COUNTRY	177	177	Australia	1

##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:

	count	unique	top	freq
CANRAT	177	2	Low	132
HDICAT	167	4	VH	59

1.3. Data Quality Assessment

Data quality findings based on assessment are as follows:

1. No duplicated rows observed.
2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
   • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
   • PATRES: Null.Count = 69, Fill.Rate = 0.610
   • ENRTER: Null.Count = 61, Fill.Rate = 0.655
   • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • GDPPER: Null.Count = 12, Fill.Rate = 0.932
   • EPISCO: Null.Count = 12, Fill.Rate = 0.932
   • HDICAT: Null.Count = 10, Fill.Rate = 0.943
   • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
   • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
   • METEMI: Null.Count = 7, Fill.Rate = 0.960
   • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
   • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
   • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
   • FORARE: Null.Count = 4, Fill.Rate = 0.977
   • TUBINC: Null.Count = 3, Fill.Rate = 0.983
   • AGRLND: Null.Count = 3, Fill.Rate = 0.983
   • POPGRO: Null.Count = 3, Fill.Rate = 0.983
   • POPDEN: Null.Count = 3, Fill.Rate = 0.983
   • URBPOP: Null.Count = 3, Fill.Rate = 0.983
   • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
   • COUNTRY=Guadeloupe: Missing.Rate= 0.909
   • COUNTRY=Martinique: Missing.Rate= 0.909
   • COUNTRY=French Guiana: Missing.Rate= 0.909
   • COUNTRY=New Caledonia: Missing.Rate= 0.500
   • COUNTRY=French Polynesia: Missing.Rate= 0.500
   • COUNTRY=Guam: Missing.Rate= 0.500
   • COUNTRY=Puerto Rico: Missing.Rate= 0.409
   • COUNTRY=North Korea: Missing.Rate= 0.227
   • COUNTRY=Somalia: Missing.Rate= 0.227
   • COUNTRY=South Sudan: Missing.Rate= 0.227
   • COUNTRY=Venezuela: Missing.Rate= 0.227
   • COUNTRY=Libya: Missing.Rate= 0.227
   • COUNTRY=Eritrea: Missing.Rate= 0.227
   • COUNTRY=Yemen: Missing.Rate= 0.227
4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
   • PM2EXP: First.Second.Mode.Ratio = 53.000
5. No low variance observed for any variable with Unique.Count.Ratio>10.
6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
   • POPDEN: Skewness = +10.267
   • GHGEMI: Skewness = +9.496
   • PATRES: Skewness = +9.284
   • METEMI: Skewness = +5.801
   • PM2EXP: Skewness = -3.141

##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()

np.int64(0)

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	COUNTRY	object	177	177	0	1.000000
1	CANRAT	category	177	177	0	1.000000
2	GDPPER	float64	177	165	12	0.932203
3	URBPOP	float64	177	174	3	0.983051
4	PATRES	float64	177	108	69	0.610169
5	RNDGDP	float64	177	74	103	0.418079
6	POPGRO	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051
9	DTHCMD	float64	177	170	7	0.960452
10	AGRLND	float64	177	174	3	0.983051
11	GHGEMI	float64	177	170	7	0.960452
12	RELOUT	float64	177	153	24	0.864407
13	METEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
15	CO2EMI	float64	177	170	7	0.960452
16	PM2EXP	float64	177	167	10	0.943503
17	POPDEN	float64	177	174	3	0.983051
18	ENRTER	float64	177	116	61	0.655367
19	GDPCAP	float64	177	170	7	0.960452
20	HDICAT	category	177	167	10	0.943503
21	EPISCO	float64	177	165	12	0.932203

##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])

20

##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
5	RNDGDP	float64	177	74	103	0.418079
4	PATRES	float64	177	108	69	0.610169
18	ENRTER	float64	177	116	61	0.655367
12	RELOUT	float64	177	153	24	0.864407
21	EPISCO	float64	177	165	12	0.932203
2	GDPPER	float64	177	165	12	0.932203
16	PM2EXP	float64	177	167	10	0.943503
20	HDICAT	category	177	167	10	0.943503
15	CO2EMI	float64	177	170	7	0.960452
13	METEMI	float64	177	170	7	0.960452
11	GHGEMI	float64	177	170	7	0.960452
9	DTHCMD	float64	177	170	7	0.960452
19	GDPCAP	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
6	POPGRO	float64	177	174	3	0.983051
3	URBPOP	float64	177	174	3	0.983051
17	POPDEN	float64	177	174	3	0.983051
10	AGRLND	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051

##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]

##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()

##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))

##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))

##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)

##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)

	Row.Name	Column.Count	Null.Count	Missing.Rate
0	Australia	22	1	0.045455
1	New Zealand	22	2	0.090909
2	Ireland	22	0	0.000000
3	United States	22	0	0.000000
4	Denmark	22	0	0.000000
...	...	...	...	...
172	Congo Republic	22	3	0.136364
173	Bhutan	22	2	0.090909
174	Nepal	22	2	0.090909
175	Gambia	22	4	0.181818
176	Niger	22	2	0.090909

177 rows × 4 columns

##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])

120

##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])

14

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))

	Row.Name	Column.Count	Null.Count	Missing.Rate
35	Guadeloupe	22	20	0.909091
39	Martinique	22	20	0.909091
56	French Guiana	22	20	0.909091
13	New Caledonia	22	11	0.500000
44	French Polynesia	22	11	0.500000
75	Guam	22	11	0.500000
53	Puerto Rico	22	9	0.409091
85	North Korea	22	6	0.272727
168	South Sudan	22	6	0.272727
132	Somalia	22	6	0.272727
117	Libya	22	5	0.227273
73	Venezuela	22	5	0.227273
161	Eritrea	22	5	0.227273
164	Yemen	22	5	0.227273

##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns

##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()

##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()

##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()

##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()

##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]

##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]

##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)

##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)

##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))

##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()

##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()

numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
0	GDPPER	1718.804896	45284.424283	34024.900890	2.346469e+05	98380.636010	77541.764380	1	1	1.000000	165	177	0.932203	1.517574	3.471992
1	URBPOP	13.345000	59.788121	61.701500	1.000000e+02	100.000000	86.699000	2	1	2.000000	173	177	0.977401	-0.210702	-0.962847
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
3	RNDGDP	0.039770	1.197474	0.873660	5.354510e+00	1.232440	3.422870	1	1	1.000000	74	177	0.418079	1.396742	1.695957
4	POPGRO	-2.079337	1.127028	1.179959	3.727101e+00	1.235701	2.204789	1	1	1.000000	174	177	0.983051	-0.195161	-0.423580
5	LIFEXP	52.777000	71.746113	72.464610	8.456000e+01	83.200000	82.256098	1	1	1.000000	174	177	0.983051	-0.357965	-0.649601
6	TUBINC	0.770000	105.005862	44.500000	5.920000e+02	12.000000	4.100000	4	3	1.333333	131	177	0.740113	1.746333	2.429368
7	DTHCMD	1.283611	21.260521	12.456279	6.520789e+01	4.941054	4.354730	1	1	1.000000	170	177	0.960452	0.900509	-0.691541
8	AGRLND	0.512821	38.793456	40.386649	8.084112e+01	46.252480	38.562911	1	1	1.000000	174	177	0.983051	0.074000	-0.926249
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	80158.025830	1	1	1.000000	170	177	0.960452	9.496120	101.637308
10	RELOUT	0.000296	39.760036	32.381668	1.000000e+02	100.000000	80.081439	3	1	3.000000	151	177	0.853107	0.501088	-0.981774
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	32241.937000	1	1	1.000000	170	177	0.960452	5.801014	38.661386
12	FORARE	0.008078	32.218177	31.509048	9.741212e+01	17.421315	37.570126	1	1	1.000000	173	177	0.977401	0.519277	-0.322589
13	CO2EMI	0.032585	3.751097	2.298368	3.172684e+01	14.772658	6.160799	1	1	1.000000	170	177	0.960452	2.721552	10.311574
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	19.331586	1	1	1.000000	174	177	0.983051	10.267750	119.995256
16	ENRTER	2.432581	49.994997	53.392460	1.433107e+02	110.139221	75.734833	1	1	1.000000	116	177	0.655367	0.275863	-0.392895
17	GDPCAP	216.827417	13992.095610	5348.192875	1.173705e+05	51722.069000	41760.594780	1	1	1.000000	170	177	0.960452	2.258568	5.938690
18	EPISCO	18.900000	42.946667	40.900000	7.790000e+01	29.600000	43.600000	3	3	1.000000	137	177	0.774011	0.641799	0.035208

##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])

1

##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
14	PM2EXP	0.274092	91.940595	100.0	100.0	100.0	100.0	106	2	53.0	61	177	0.344633	-3.141557	9.032386

##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])

5

##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	19.331586	1	1	1.000000	174	177	0.983051	10.267750	119.995256
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	80158.025830	1	1	1.000000	170	177	0.960452	9.496120	101.637308
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	32241.937000	1	1	1.000000	170	177	0.960452	5.801014	38.661386
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386

##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')

##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns

##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]

##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]

##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]

##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]

##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)

##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)

##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))

##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)

object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)

	Object.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	COUNTRY	Australia	New Zealand	1	1	1.0	177	177	1.0

##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')

##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns

##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]

##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]

##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)

##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)

##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)

categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)

	Categorical.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	CANRAT	Low	High	132	45	2.933333	2	177	0.011299
1	HDICAT	VH	H	59	39	1.512821	4	177	0.022599

##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])

0

1.4. Data Preprocessing

1.4.1 Data Cleaning

1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
   • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
     • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
     • PATRES: Null.Count = 69, Fill.Rate = 0.610
     • ENRTER: Null.Count = 61, Fill.Rate = 0.655
     • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
     • COUNTRY=Guadeloupe: Missing.Rate= 0.909
     • COUNTRY=Martinique: Missing.Rate= 0.909
     • COUNTRY=French Guiana: Missing.Rate= 0.909
     • COUNTRY=New Caledonia: Missing.Rate= 0.500
     • COUNTRY=French Polynesia: Missing.Rate= 0.500
     • COUNTRY=Guam: Missing.Rate= 0.500
     • COUNTRY=Puerto Rico: Missing.Rate= 0.409
     • COUNTRY=North Korea: Missing.Rate= 0.227
     • COUNTRY=Somalia: Missing.Rate= 0.227
     • COUNTRY=South Sudan: Missing.Rate= 0.227
     • COUNTRY=Venezuela: Missing.Rate= 0.227
     • COUNTRY=Libya: Missing.Rate= 0.227
     • COUNTRY=Eritrea: Missing.Rate= 0.227
     • COUNTRY=Yemen: Missing.Rate= 0.227
2. No variables were removed due to zero or near-zero variance.
3. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 15/18 predictor (numeric)
       • GDPPER
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/18 predictor (categorical)
       • HDICAT

##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)

##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)

##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.4.2 Missing Data Imputation

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
   • GDPPER: Null.Count = 1
   • FORARE: Null.Count = 1
   • PM2EXP: Null.Count = 5
2. Missing data for categorical variables were imputed using the most frequent value.
   • HDICAP: Null.Count = 1

##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count
0	COUNTRY	object	163	163	0
1	CANRAT	category	163	163	0
2	GDPPER	float64	163	162	1
3	URBPOP	float64	163	163	0
4	POPGRO	float64	163	163	0
5	LIFEXP	float64	163	163	0
6	TUBINC	float64	163	163	0
7	DTHCMD	float64	163	163	0
8	AGRLND	float64	163	163	0
9	GHGEMI	float64	163	163	0
10	METEMI	float64	163	163	0
11	FORARE	float64	163	162	1
12	CO2EMI	float64	163	163	0
13	PM2EXP	float64	163	158	5
14	POPDEN	float64	163	163	0
15	GDPCAP	float64	163	163	0
16	HDICAT	category	163	162	1
17	EPISCO	float64	163	163	0

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')

##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')

##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	NaN	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()

##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)

##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)

##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)

##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	65.867296	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')

##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'] = cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0])
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)

##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')  

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	GDPPER	float64	163	163	0	1.0
1	URBPOP	float64	163	163	0	1.0
2	POPGRO	float64	163	163	0	1.0
3	LIFEXP	float64	163	163	0	1.0
4	TUBINC	float64	163	163	0	1.0
5	DTHCMD	float64	163	163	0	1.0
6	AGRLND	float64	163	163	0	1.0
7	GHGEMI	float64	163	163	0	1.0
8	METEMI	float64	163	163	0	1.0
9	FORARE	float64	163	163	0	1.0
10	CO2EMI	float64	163	163	0	1.0
11	PM2EXP	float64	163	163	0	1.0
12	POPDEN	float64	163	163	0	1.0
13	GDPCAP	float64	163	163	0	1.0
14	EPISCO	float64	163	163	0	1.0
15	CANRAT	category	163	163	0	1.0
16	HDICAT	category	163	163	0	1.0

1.4.3 Outlier Detection

1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
   • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
   • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
   • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
   • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
   • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
   • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
   • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
   • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
   • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
   • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910

##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()

##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1

##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()

##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)

##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)

	Numeric.Column.Name	Skewness	Outlier.Count	Row.Count	Outlier.Ratio
0	GDPPER	1.554457	3	163	0.018405
1	URBPOP	-0.212327	0	163	0.000000
2	POPGRO	-0.181666	0	163	0.000000
3	LIFEXP	-0.329704	0	163	0.000000
4	TUBINC	1.747962	12	163	0.073620
5	DTHCMD	0.930709	0	163	0.000000
6	AGRLND	0.035315	0	163	0.000000
7	GHGEMI	9.299960	27	163	0.165644
8	METEMI	5.688689	20	163	0.122699
9	FORARE	0.563015	0	163	0.000000
10	CO2EMI	2.693585	11	163	0.067485
11	PM2EXP	-3.088403	37	163	0.226994
12	POPDEN	9.972806	20	163	0.122699
13	GDPCAP	2.311079	22	163	0.134969
14	EPISCO	0.635994	3	163	0.018405

##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)

1.4.4 Collinearity

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
   • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
   • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
   • GDPPER: Pearson.Correlation.Coefficient = +0.690
   • METEMI: Pearson.Correlation.Coefficient = +0.062
4. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 16 columns (variables)
     • 1/16 metadata (object)
       • COUNTRY
     • 1/16 target (categorical)
       • CANRAT
     • 13/16 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/16 predictor (categorical)
       • HDICAT

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})  

##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])

##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))

	Pearson.Correlation.Coefficient	Correlation.PValue
GDPPER_GDPCAP	0.921010	8.158179e-68
GHGEMI_METEMI	0.905121	1.087643e-61
POPGRO_DTHCMD	0.759470	7.124695e-32
GDPPER_LIFEXP	0.755787	2.055178e-31
GDPCAP_EPISCO	0.696707	5.312642e-25
LIFEXP_GDPCAP	0.683834	8.321371e-24
GDPPER_EPISCO	0.680812	1.555304e-23
GDPPER_URBPOP	0.666394	2.781623e-22
GDPPER_CO2EMI	0.654958	2.450029e-21
TUBINC_DTHCMD	0.643615	1.936081e-20
URBPOP_LIFEXP	0.623997	5.669778e-19
LIFEXP_EPISCO	0.620271	1.048393e-18
URBPOP_GDPCAP	0.559181	8.624533e-15
CO2EMI_GDPCAP	0.550221	2.782997e-14
URBPOP_CO2EMI	0.550046	2.846393e-14
LIFEXP_CO2EMI	0.531305	2.951829e-13
URBPOP_EPISCO	0.510131	3.507463e-12
POPGRO_TUBINC	0.442339	3.384403e-09
DTHCMD_PM2EXP	0.283199	2.491837e-04
CO2EMI_EPISCO	0.282734	2.553620e-04

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)  

##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.4.5 Shape Transformation

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
3. One variable which remained skewed even after applying shape transformation was removed.
   • PM2EXP
4. The transformed dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)

##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)

##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 12)

1.4.6 Centering and Scaling

1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
2. The scaled dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)

##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)

1.4.7 Data Encoding

1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
   • HDICAP_L
   • HDICAP_M
   • HDICAP_H
   • HDICAP_VH

##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])

##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.4.8 Preprocessed Data Description

1. The preprocessed dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 12/18 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 4/18 predictor (categorical)
       • HDICAT_L
       • HDICAT_M
       • HDICAT_H
       • HDICAT_VH

##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')  

##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 16)

1.5. Data Exploration

1.5.1 Exploratory Data Analysis

1. Bivariate analysis identified individual predictors with generally positive association to the target variable based on visual inspection.
2. Higher values or higher proportions for the following predictors are associated with the CANRAT HIGH category:
   • URBPOP
   • LIFEXP
   • CO2EMI
   • GDPCAP
   • EPISCO
   • HDICAP_VH=1
3. Decreasing values or smaller proportions for the following predictors are associated with the CANRAT LOW category:
   • POPGRO
   • TUBINC
   • DTHCMD
   • HDICAP_L=0
   • HDICAP_M=0
   • HDICAP_H=0
4. Values for the following predictors are not associated with the CANRAT HIGH or LOW categories:
   • AGRLND
   • GHGEMI
   • FORARE
   • POPDEN

##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1) 
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)

##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable, observed=True)])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(y_variable)
    ax.set_ylabel(x_variable)
    ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
    ax = axes[i]
    category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable], observed=True).size().unstack(fill_value=0)
    category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
    category_proportions.plot(kind='bar', stacked=True, ax=ax)
    ax.set_title(f'{x_variable} Versus {y_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel('Proportions')

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

1.5.2 Hypothesis Testing

1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Difference in the means between groups LOW and HIGH is equal to zero
   • Alternative: Difference in the means between groups LOW and HIGH is not equal to zero
2. There is sufficient evidence to conclude of a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable in 9 of the 12 numeric predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
   • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, Correlation.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, Correlation.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, Correlation.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, Correlation.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, Correlation.PValue=0.000
   • POPGRO: T.Test.Statistic=+4.905, Correlation.PValue=0.000
   • GHGEMI: T.Test.Statistic=-2.243, Correlation.PValue=0.026
3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: The categorical predictor is independent of the categorical target variable
   • Alternative: The categorical predictor is dependent of the categorical target variable
4. There is sufficient evidence to conclude of a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable in all 4 categorical predictors given their high chisquare statistic values with reported low p-values less than the significance level of 0.05.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
   • HDICAT_H: ChiSquare.Test.Statistic=13.860, ChiSquare.Test.PValue=0.000
   • HDICAT_M: ChiSquare.Test.Statistic=10.286, ChiSquare.Test.PValue=0.001
   • HDICAT_L: ChiSquare.Test.Statistic=9.081, ChiSquare.Test.PValue=0.002

##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
    group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
    cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
        group_0[numeric_column], 
        group_1[numeric_column], 
        equal_var=True)

##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))

	T.Test.Statistic	T.Test.PValue
CANRAT_GDPCAP	-11.936988	6.247937e-24
CANRAT_EPISCO	-11.788870	1.605980e-23
CANRAT_LIFEXP	-10.979098	2.754214e-21
CANRAT_TUBINC	9.608760	1.463678e-17
CANRAT_DTHCMD	8.375558	2.552108e-14
CANRAT_CO2EMI	-7.030702	5.537463e-11
CANRAT_URBPOP	-6.541001	7.734940e-10
CANRAT_POPGRO	4.904817	2.269446e-06
CANRAT_GHGEMI	-2.243089	2.625563e-02
CANRAT_FORARE	-1.174143	2.420717e-01
CANRAT_POPDEN	-0.495221	6.211191e-01
CANRAT_AGRLND	-0.047628	9.620720e-01

##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column], 
                                    cancer_rate_preprocessed_categorical['CANRAT'])
    cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
        contingency_table)[0:2]

##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))

	ChiSquare.Test.Statistic	ChiSquare.Test.PValue
CANRAT_HDICAT_VH	76.764134	1.926446e-18
CANRAT_HDICAT_M	13.860367	1.969074e-04
CANRAT_HDICAT_L	10.285575	1.340742e-03
CANRAT_HDICAT_H	9.080788	2.583087e-03

1.6. Neural Network Classification Gradient and Weight Updates

1.6.1 Premodelling Data Description

1. Among the predictor variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable, only 2 were retained with the highest absolute t-test statistic values with reported low p-values less than the significance level of 0.05..
   • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000

##################################
# Filtering certain numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_combined.drop(['URBPOP', 'POPGRO', 'LIFEXP', 'TUBINC', 'DTHCMD', 'AGRLND', 'GHGEMI','FORARE', 'CO2EMI', 'POPDEN'], axis=1)
cancer_rate_premodelling.columns

Index(['GDPCAP', 'EPISCO', 'CANRAT'], dtype='object')

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 3)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

GDPCAP     float64
EPISCO     float64
CANRAT    category
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate_premodelling.head()

	GDPCAP	EPISCO	CANRAT
0	1.549766	1.306738	High
1	1.407752	1.102912	High
2	1.879374	1.145832	High
3	1.685426	0.739753	High
4	1.657777	2.218327	High

##################################
# Converting the dataframe to
# a numpy array
##################################
cancer_rate_premodelling_matrix = cancer_rate_premodelling.to_numpy()

##################################
# Formulating the scatterplot
# of the selected numeric predictors
# by categorical response classes
##################################
fig, ax = plt.subplots(figsize=(7, 7))
ax.plot(cancer_rate_premodelling_matrix[cancer_rate_premodelling_matrix[:,2]=='High', 0],
        cancer_rate_premodelling_matrix[cancer_rate_premodelling_matrix[:,2]=='High', 1], 
        'o', 
        label='High', 
        color='darkslateblue')
ax.plot(cancer_rate_premodelling_matrix[cancer_rate_premodelling_matrix[:,2]=='Low', 0],
        cancer_rate_premodelling_matrix[cancer_rate_premodelling_matrix[:,2]=='Low', 1], 
        'x', 
        label='Low', 
        color='chocolate')
ax.axes.set_ylabel('EPISCO')
ax.axes.set_xlabel('GDPCAP')
ax.set_xlim(-3,3)
ax.set_ylim(-3,3)
ax.set(title='CANRAT Class Distribution')
ax.legend(loc='upper left',title='CANRAT');

##################################
# Preparing the data and
# and converting to a suitable format
# as a neural network model input
##################################
matrix_x_values = cancer_rate_premodelling.iloc[:,0:2].to_numpy()
y_values = np.where(cancer_rate_premodelling['CANRAT'] == 'High', 1, 0)

1.6.2 Sigmoid Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Sigmoid Activation Function transforms a variable by determining the quotient between one and the sum of one and the Euler's constant raised to the given value of the variable. The resulting output resembles a smooth S-shaped curve which ranges from zero to one. Since, it squashes the input values between zero and one, it is useful for binary classification. The curve profile is not zero-centered which can lead to vanishing gradients problem during backpropagation. It is also prone to saturation, causing gradients to vanish when the input is too large or too small.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Sigmoid Activation Function
3. The final loss estimate determined as 0.27805 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a Sigmoid activation function, the neural network model performance is estimated as follows:
   • Accuracy = 74.84662
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a Sigmoid activation function was not optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training

##################################
# Creating a class object
# for the neural network algorithm
# using a sigmoid activation function
##################################
class NeuralNetwork_Sigmoid:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def sigmoid(self, x):
        return 1 / (1 + np.exp(-x))
    
    def sigmoid_derivative(self, x):
        return x * (1 - x)
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.sigmoid(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.sigmoid(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.sigmoid(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
       # Computing the output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing the hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * self.sigmoid_derivative(self.hidden3)
        
        # Computing the hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * self.sigmoid_derivative(self.hidden2)
        
        # Computing the hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * self.sigmoid_derivative(self.hidden1)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing the updated gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
                       
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with sigmoid activation function
##################################
np.random.seed(88888)
nn_sigmoid = NeuralNetwork_Sigmoid(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with sigmoid activation function
##################################
nn_sigmoid.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 0.316009360251368, Accuracy 0.7484662576687117
Epoch 100: Loss 0.29042169855505795, Accuracy 0.7484662576687117
Epoch 200: Loss 0.28610782259171613, Accuracy 0.7484662576687117
Epoch 300: Loss 0.2846113282316412, Accuracy 0.7484662576687117
Epoch 400: Loss 0.2835652925120681, Accuracy 0.7484662576687117
Epoch 500: Loss 0.2826104420839403, Accuracy 0.7484662576687117
Epoch 600: Loss 0.281684171341873, Accuracy 0.7484662576687117
Epoch 700: Loss 0.28077152406724765, Accuracy 0.7484662576687117
Epoch 800: Loss 0.2798656430150373, Accuracy 0.7484662576687117
Epoch 900: Loss 0.2789612062764938, Accuracy 0.7484662576687117
Epoch 1000: Loss 0.27805328141665736, Accuracy 0.7484662576687117

##################################
# Plotting the computed gradients
# of the neural network model
# with sigmoid activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_sigmoid.gradients.items():
    plt.plot(value, label=key)
plt.title('Sigmoid Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with sigmoid activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_sigmoid.losses)
plt.title('Sigmoid Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with sigmoid activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_sigmoid.accuracies)
plt.title('Sigmoid Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
##################################
sigmoid_metrics = pd.DataFrame(["ACCURACY","LOSS"])
sigmoid_values = pd.DataFrame([nn_sigmoid.accuracies[-1],nn_sigmoid.losses[-1]])
sigmoid_method = pd.DataFrame(["Sigmoid"]*2)
sigmoid_summary = pd.concat([sigmoid_metrics, 
                             sigmoid_values,
                             sigmoid_method], axis=1)
sigmoid_summary.columns = ['Metric', 'Value', 'Method']
sigmoid_summary.reset_index(inplace=True, drop=True)
display(sigmoid_summary)

	Metric	Value	Method
0	ACCURACY	0.748466	Sigmoid
1	LOSS	0.278053	Sigmoid

1.6.3 Rectified Linear Unit Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data. This activation function is commonly used in the output layer for binary classification problems.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Rectified Linear Unit Activation Function transforms a variable by determining the maximum between zero and the given value of the variable. The resulting output which ranges from zero to positive infinity is a piecewise linear function that returns zero for negative inputs and the input value for positive inputs. It is a simple and computationally efficient method which solves the vanishing gradients problem and accelerates convergence by avoiding saturation to positive values. However, it can suffer from dying RELU problem where neurons become inactive (outputs zero) for all inputs during training if large gradients consistently flow through them. The curve profile is not zero-centered which can lead to vanishing gradients problem during backpropagation. It is also prone to saturation, causing gradients to vanish when the input is too large or too small. This activation function is widely used in hidden layers of deep neural networks.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Rectified Linear Unit Activation Function (RELU)
3. The final loss estimate determined as 0.09623 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a RELU activation function, the neural network model performance is estimated as follows:
   • Accuracy = 93.25153
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a RELU activation function was optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training

##################################
# Creating a class object
# for the neural network algorithm
# using a RELU activation function
##################################
import numpy as np
import matplotlib.pyplot as plt

class NeuralNetwork_RELU:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def relu(self, x):
        return np.maximum(0, x)
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.relu(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.relu(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.relu(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
        # Computing the output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing the hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * (self.hidden3 > 0)
        
        # Computing the hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * (self.hidden2 > 0)
        
        # Computing the hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * (self.hidden1 > 0)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing computed gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
        
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with RELU activation function
##################################
np.random.seed(88888)
nn_relu = NeuralNetwork_RELU(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with RELU activation function
##################################
nn_relu.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 0.3700211712894535, Accuracy 0.2822085889570552
Epoch 100: Loss 0.17942495875926145, Accuracy 0.901840490797546
Epoch 200: Loss 0.13224275039852637, Accuracy 0.9079754601226994
Epoch 300: Loss 0.11676788805888334, Accuracy 0.9141104294478528
Epoch 400: Loss 0.10890482140743656, Accuracy 0.9202453987730062
Epoch 500: Loss 0.10455617528678197, Accuracy 0.9202453987730062
Epoch 600: Loss 0.10189142339749103, Accuracy 0.9202453987730062
Epoch 700: Loss 0.09987241742776296, Accuracy 0.9263803680981595
Epoch 800: Loss 0.09805768178547355, Accuracy 0.9325153374233128
Epoch 900: Loss 0.09697282986650113, Accuracy 0.9325153374233128
Epoch 1000: Loss 0.0962368489065675, Accuracy 0.9325153374233128

##################################
# Plotting the computed gradients
# of the neural network model
# with RELU activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_relu.gradients.items():
    plt.plot(value, label=key)
plt.title('RELU Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with RELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_relu.losses)
plt.title('RELU Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with RELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_relu.accuracies)
plt.title('RELU Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
# using a RELU activation function
##################################
relu_metrics = pd.DataFrame(["ACCURACY","LOSS"])
relu_values = pd.DataFrame([nn_relu.accuracies[-1],nn_relu.losses[-1]])
relu_method = pd.DataFrame(["RELU"]*2)
relu_summary = pd.concat([relu_metrics, 
                          relu_values,
                          relu_method], axis=1)
relu_summary.columns = ['Metric', 'Value', 'Method']
relu_summary.reset_index(inplace=True, drop=True)
display(relu_summary)

	Metric	Value	Method
0	ACCURACY	0.932515	RELU
1	LOSS	0.096237	RELU

1.6.4 Leaky Rectified Linear Unit Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Leaky Rectified Linear Unit Activation Function transforms a variable by using the given value of the variable when it is greater than zero, but multiplies this value with a specific alpha otherwise. The resulting output which ranges from negative infinity to positive infinity is a piecewise linear function that returns the input value for positive inputs but allows a small non-zero value when the input is negative, preventing the dying RELU problem. It introduces a small slope for negative inputs, keeping the gradient flowing during backpropagation. This activation function provides a good alternative to RELU especially in networks where dying RELU is a concern.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Leaky Rectified Linear Unit Activation Function (Leaky RELU)
3. The final loss estimate determined as 0.09377 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a Leaky RELU activation function, the neural network model performance is estimated as follows:
   • Accuracy = 92.63803
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a Leaky RELU activation function was optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training.

##################################
# Creating a class object
# for the neural network algorithm
# using a Leaky RELU activation function
##################################
class NeuralNetwork_LeakyRELU:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def leaky_relu(self, x, alpha=0.10):
        return np.where(x > 0, x, alpha * x)
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.leaky_relu(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.leaky_relu(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.leaky_relu(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
        # Computing the output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing the hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * (self.hidden3 > 0) + 0.01 * hidden3_error * (self.hidden3 <= 0)
        
        # Computing the hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * (self.hidden2 > 0) + 0.01 * hidden2_error * (self.hidden2 <= 0)
        
        # Computing the hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * (self.hidden1 > 0) + 0.01 * hidden1_error * (self.hidden1 <= 0)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing updated gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
        
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)
    

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with RELU activation function
##################################
np.random.seed(88888)
nn_leakyrelu = NeuralNetwork_LeakyRELU(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with Leaky RELU activation function
##################################
nn_leakyrelu.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 0.5521727210322692, Accuracy 0.3803680981595092
Epoch 100: Loss 0.15843767165873718, Accuracy 0.9079754601226994
Epoch 200: Loss 0.12183916598907374, Accuracy 0.901840490797546
Epoch 300: Loss 0.11001101254184709, Accuracy 0.9263803680981595
Epoch 400: Loss 0.10402068587112542, Accuracy 0.9263803680981595
Epoch 500: Loss 0.10064359626697073, Accuracy 0.9263803680981595
Epoch 600: Loss 0.09839877372876209, Accuracy 0.9325153374233128
Epoch 700: Loss 0.09675845212053226, Accuracy 0.9325153374233128
Epoch 800: Loss 0.09550172181604445, Accuracy 0.9263803680981595
Epoch 900: Loss 0.09454243365295599, Accuracy 0.9263803680981595
Epoch 1000: Loss 0.09377715998492316, Accuracy 0.9263803680981595

##################################
# Plotting the computed gradients
# of the neural network model
# with RELU activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_leakyrelu.gradients.items():
    plt.plot(value, label=key)
plt.title('Leaky RELU Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with Leaky RELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_leakyrelu.losses)
plt.title('Leaky RELU Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with Leaky RELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_leakyrelu.accuracies)
plt.title('Leaky RELU Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
# using a Leaky RELU activation function
##################################
leakyrelu_metrics = pd.DataFrame(["ACCURACY","LOSS"])
leakyrelu_values = pd.DataFrame([nn_leakyrelu.accuracies[-1],nn_leakyrelu.losses[-1]])
leakyrelu_method = pd.DataFrame(["Leaky_RELU"]*2)
leakyrelu_summary = pd.concat([leakyrelu_metrics,
                               leakyrelu_values,
                               leakyrelu_method], axis=1)
leakyrelu_summary.columns = ['Metric', 'Value', 'Method']
leakyrelu_summary.reset_index(inplace=True, drop=True)
display(leakyrelu_summary)

	Metric	Value	Method
0	ACCURACY	0.926380	Leaky_RELU
1	LOSS	0.093777	Leaky_RELU

1.6.5 Exponential Linear Unit Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Exponential Linear Unit Activation Function transforms a variable by using the given value of the variable when it is greater than zero, but uses this value as power for the Euler's constant then subtracted by one prior to multiplying with a specific alpha otherwise. The resulting output which ranges from negative infinity to positive infinity is similar to a Leaky RELU profile but with an exponential function for negative inputs, allowing negative values to have negative outputs. Its profile is smooth and differentiable helping speed up convergence which can lead to faster learning. This activation function can be a good choice when smoother and faster convergence are desired.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Exponential Linear Unit (ELU)
3. The final loss estimate determined as 0.09443 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a ELU activation function, the neural network model performance is estimated as follows:
   • Accuracy = 92.63803
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a ELU activation function was optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training.

##################################
# Creating a class object
# for the neural network algorithm
# using an ELU activation function
##################################
class NeuralNetwork_ELU:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def elu(self, x, alpha=1.0):
        return np.where(x > 0, x, alpha * (np.exp(x) - 1))
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.elu(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.elu(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.elu(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
        # Computing output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * (self.hidden3 > 0) + (np.exp(self.hidden3) - 1) * (self.hidden3 <= 0)
        
        # Computing hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * (self.hidden2 > 0) + (np.exp(self.hidden2) - 1) * (self.hidden2 <= 0)
        
        # Computing hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * (self.hidden1 > 0) + (np.exp(self.hidden1) - 1) * (self.hidden1 <= 0)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing updated gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
        
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)
    

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with ELU activation function
##################################
np.random.seed(88888)
nn_elu = NeuralNetwork_ELU(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with ELU activation function
##################################
nn_elu.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 0.6239532743548977, Accuracy 0.6748466257668712
Epoch 100: Loss 0.18559850810160425, Accuracy 0.7852760736196319
Epoch 200: Loss 0.155914841757359, Accuracy 0.8711656441717791
Epoch 300: Loss 0.13187741248142648, Accuracy 0.901840490797546
Epoch 400: Loss 0.11492254996431339, Accuracy 0.9141104294478528
Epoch 500: Loss 0.10509129045149054, Accuracy 0.9263803680981595
Epoch 600: Loss 0.1002135530989771, Accuracy 0.9263803680981595
Epoch 700: Loss 0.09771434144678748, Accuracy 0.9263803680981595
Epoch 800: Loss 0.09611887321358306, Accuracy 0.9263803680981595
Epoch 900: Loss 0.0950962862639301, Accuracy 0.9263803680981595
Epoch 1000: Loss 0.09443192186232496, Accuracy 0.9263803680981595

##################################
# Plotting the computed gradients
# of the neural network model
# with ELU activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_elu.gradients.items():
    plt.plot(value, label=key)
plt.title('ELU Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with ELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_elu.losses)
plt.title('ELU Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with ELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_elu.accuracies)
plt.title('ELU Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
# using a ELU activation function
##################################
elu_metrics = pd.DataFrame(["ACCURACY","LOSS"])
elu_values = pd.DataFrame([nn_elu.accuracies[-1],nn_elu.losses[-1]])
elu_method = pd.DataFrame(["ELU"]*2)
elu_summary = pd.concat([elu_metrics,
                         elu_values,
                         elu_method], axis=1)
elu_summary.columns = ['Metric', 'Value', 'Method']
elu_summary.reset_index(inplace=True, drop=True)
display(elu_summary)

	Metric	Value	Method
0	ACCURACY	0.926380	ELU
1	LOSS	0.094432	ELU

1.6.6 Scaled Exponential Linear Unit Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Scaled Exponential Linear Unit Activation Function transforms a variable by using the given value of the variable when it is greater than zero, but uses this value as power for the Euler's constant then subtracted by one prior to multiplying with a specific alpha otherwise - but multiplying either value with a separately defined scaling factor. The resulting output which ranges from negative infinity to positive infinity is similar to an ELU profile but was designed to maintain a stable mean and variance of activations across layers, promoting self-normalization. This method provides a solution to vanishing and exploding gradient problems in deep networks. This activation function is typically used in deep neural networks where normalizing the activations is crucial for stability.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Scaled Exponential Linear Unit (SELU)
3. The final loss estimate determined as 0.09674 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a SELU activation function, the neural network model performance is estimated as follows:
   • Accuracy = 90.18404
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a SELU activation function was optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training.

##################################
# Creating a class object
# for the neural network algorithm
# using a SELU activation function
##################################
class NeuralNetwork_SELU:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def selu(self, x):
        alpha = 1.6732632423543772848170429916717
        scale = 1.0507009873554804934193349852946
        return scale * np.where(x > 0, x, alpha * (np.exp(x) - 1))
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.selu(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.selu(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.selu(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
        # Computing output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * (self.hidden3 > 0)
        
        # Computing hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * (self.hidden2 > 0)
        
        # Computing hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * (self.hidden1 > 0)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing computed gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
        
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with SELU activation function
##################################
np.random.seed(88888)
nn_selu = NeuralNetwork_SELU(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with SELU activation function
##################################
nn_selu.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 1.0981323160350687, Accuracy 0.3006134969325153
Epoch 100: Loss 0.1944859567995401, Accuracy 0.8343558282208589
Epoch 200: Loss 0.15078198879619664, Accuracy 0.8773006134969326
Epoch 300: Loss 0.13087303505062367, Accuracy 0.8773006134969326
Epoch 400: Loss 0.11912606258714767, Accuracy 0.8773006134969326
Epoch 500: Loss 0.11149519060605416, Accuracy 0.8895705521472392
Epoch 600: Loss 0.10641661771976164, Accuracy 0.901840490797546
Epoch 700: Loss 0.10280038529399513, Accuracy 0.901840490797546
Epoch 800: Loss 0.1001744396601595, Accuracy 0.901840490797546
Epoch 900: Loss 0.09822970379617403, Accuracy 0.8957055214723927
Epoch 1000: Loss 0.09674886443517218, Accuracy 0.901840490797546

##################################
# Plotting the computed gradients
# of the neural network model
# with SELU activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_selu.gradients.items():
    plt.plot(value, label=key)
plt.title('SELU Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with SELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_selu.losses)
plt.title('SELU Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with SELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_selu.accuracies)
plt.title('SELU Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
# using a SELU activation function
##################################
selu_metrics = pd.DataFrame(["ACCURACY","LOSS"])
selu_values = pd.DataFrame([nn_selu.accuracies[-1],nn_selu.losses[-1]])
selu_method = pd.DataFrame(["SELU"]*2)
selu_summary = pd.concat([selu_metrics,
                          selu_values,
                          selu_method], axis=1)
selu_summary.columns = ['Metric', 'Value', 'Method']
selu_summary.reset_index(inplace=True, drop=True)
display(selu_summary)

	Metric	Value	Method
0	ACCURACY	0.901840	SELU
1	LOSS	0.096749	SELU

1.6.7 Randomized Leaky Rectified Linear Unit Activation Function

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Activation Functions play a crucial role in neural networks by introducing non-linearity into the network, enabling the model to learn complex patterns and relationships within the data. In the context of a neural network classification model, activation functions are applied to the output of each neuron in the hidden layers to introduce non-linear mappings between the input and output, allowing the network to approximate complex functions and make non-linear decisions. Activation functions are significant during model development by introducing non-linearity (without activation functions, the neural network would simply be a series of linear transformations, no matter how many layers it has. Activation functions introduce non-linearities to the model, enabling it to learn and represent complex patterns and relationships in the data); propagating back gradients (activation functions help in the backpropagation algorithm by providing gradients that indicate the direction and magnitude of adjustments to the weights during training. These gradients are necessary for optimizing the network's parameters through techniques like gradient descent}; and normalizing outputs (activation functions also help in normalizing the output of each neuron, ensuring that it falls within a specific range. This normalization prevents the activation values from becoming too large or too small, which can lead to numerical instability or saturation of gradients during training). The choice of activation function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

Randomized Leaky Rectified Linear Unit Activation Function transforms a variable by using the given value of the variable when it is greater than zero, but multiplies this value with a random number obtained from a uniform distribution. The resulting output which ranges from negative infinity to positive infinity is similar to a Leaky RELU profile but the slope parameter is randomly initialized and updated during training. This process introduces randomness into the activations, acting as a form of regularization and reducing overfitting. This provides a trade-off between the benefits of RELU and Leaky RELU along with the regularization effect of randomness. This activation function is typically useful when dealing with overfitting or training large neural networks where regularization is necessary.

1. A neural network with the following structure was formulated:
   • Hidden Layer = 3
   • Number of Nodes per Hidden Layer = 4
2. The backpropagation and gradient descent algorithms were implemented with parameter settings described as follows:
   • Learning Rate = 0.01
   • Epochs = 1000
   • Activation Function = Randomized Leaky Rectified Linear Unit (RReLU)
3. The final loss estimate determined as 0.09765 at the 1000th epoch was not optimally low as compared to those obtained using the other parameter settings.
4. Applying the backpropagation and gradient descent algorithms with a RReLU activation function, the neural network model performance is estimated as follows:
   • Accuracy = 92.63803
5. The estimated classification accuracy using the backpropagation and gradient descent algorithms with a RReLU activation function was optimal as compared to those obtained using the other parameter settings, also demonstrating a consistently smooth profile during the epoch training.

##################################
# Creating a class object
# for the neural network algorithm
# using a RRELU activation function
##################################
class NeuralNetwork_RRELU:
    def __init__(self, input_size, hidden_size1, hidden_size2, hidden_size3, output_size):
        self.weights1 = np.random.randn(input_size, hidden_size1)
        self.bias1 = np.zeros((1, hidden_size1))
        
        self.weights2 = np.random.randn(hidden_size1, hidden_size2)
        self.bias2 = np.zeros((1, hidden_size2))
        
        self.weights3 = np.random.randn(hidden_size2, hidden_size3)
        self.bias3 = np.zeros((1, hidden_size3))
        
        self.weights4 = np.random.randn(hidden_size3, output_size)
        self.bias4 = np.zeros((1, output_size))
        
        self.gradients = {'dw1': [], 'db1': [], 'dw2': [], 'db2': [], 'dw3': [], 'db3': [], 'dw4': [], 'db4': []}
        self.losses = []
        self.accuracies = []
        
    def randomized_leaky_relu(self, x):
        alpha = np.random.uniform(0.01, 0.1)  # Randomly generate alpha in the range [0.01, 0.1]
        return np.where(x > 0, x, alpha * x)
    
    def softmax(self, x):
        exps = np.exp(x - np.max(x, axis=1, keepdims=True))
        return exps / np.sum(exps, axis=1, keepdims=True)
    
    def forward(self, x):
        self.hidden1 = self.randomized_leaky_relu(np.dot(x, self.weights1) + self.bias1)
        self.hidden2 = self.randomized_leaky_relu(np.dot(self.hidden1, self.weights2) + self.bias2)
        self.hidden3 = self.randomized_leaky_relu(np.dot(self.hidden2, self.weights3) + self.bias3)
        self.output = self.softmax(np.dot(self.hidden3, self.weights4) + self.bias4)
        
        return self.output
    
    def backward(self, x, y, lr):
        m = x.shape[0]
        
        # Computing output layer gradients
        output_error = self.output - y
        output_delta = output_error / m
        
        # Computing hidden layer 3 gradients
        hidden3_error = np.dot(output_delta, self.weights4.T)
        hidden3_delta = hidden3_error * (self.hidden3 > 0) + 0.01 * hidden3_error * (self.hidden3 <= 0)
        
        # Computing hidden layer 2 gradients
        hidden2_error = np.dot(hidden3_delta, self.weights3.T)
        hidden2_delta = hidden2_error * (self.hidden2 > 0) + 0.01 * hidden2_error * (self.hidden2 <= 0)
        
        # Computing hidden layer 1 gradients
        hidden1_error = np.dot(hidden2_delta, self.weights2.T)
        hidden1_delta = hidden1_error * (self.hidden1 > 0) + 0.01 * hidden1_error * (self.hidden1 <= 0)
        
        # Updating weights and biases based on computed gradients
        self.weights4 -= lr * np.dot(self.hidden3.T, output_delta)
        self.bias4 -= lr * np.sum(output_delta, axis=0, keepdims=True)
        
        self.weights3 -= lr * np.dot(self.hidden2.T, hidden3_delta)
        self.bias3 -= lr * np.sum(hidden3_delta, axis=0, keepdims=True)
        
        self.weights2 -= lr * np.dot(self.hidden1.T, hidden2_delta)
        self.bias2 -= lr * np.sum(hidden2_delta, axis=0, keepdims=True)
        
        self.weights1 -= lr * np.dot(x.T, hidden1_delta)
        self.bias1 -= lr * np.sum(hidden1_delta, axis=0, keepdims=True)
        
        # Storing computed gradients
        self.gradients['dw1'].append(np.mean(np.abs(hidden1_delta)))
        self.gradients['db1'].append(np.mean(np.abs(self.bias1)))
        self.gradients['dw2'].append(np.mean(np.abs(hidden2_delta)))
        self.gradients['db2'].append(np.mean(np.abs(self.bias2)))
        self.gradients['dw3'].append(np.mean(np.abs(hidden3_delta)))
        self.gradients['db3'].append(np.mean(np.abs(self.bias3)))
        self.gradients['dw4'].append(np.mean(np.abs(output_delta)))
        self.gradients['db4'].append(np.mean(np.abs(self.bias4)))
        
    def train(self, x, y, epochs, lr):
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, lr)
            loss = -np.mean(y * np.log(output))
            self.losses.append(loss)
            accuracy = self.accuracy(x, np.argmax(y, axis=1))
            self.accuracies.append(accuracy)
            if i % 100 == 0:
                print(f'Epoch {i}: Loss {loss}, Accuracy {accuracy}')
                
    def predict(self, x):
        return np.argmax(self.forward(x), axis=1)
    
    def accuracy(self, x, y):
        pred = self.predict(x)
        return np.mean(pred == y)
    

##################################
# Preparing the training data
##################################
X = matrix_x_values
y = y_values

##################################
# Performing a one-hot encoding
# of the target response labels
##################################
num_classes = 2
y_one_hot = np.eye(num_classes)[y]

##################################
# Defining the neural network components
##################################
input_size = 2
hidden_size1 = 4
hidden_size2 = 4
hidden_size3 = 4
output_size = 2

##################################
# Initializing a neural network model object
# with RRELU activation function
##################################
np.random.seed(88888)
nn_rrelu = NeuralNetwork_RRELU(input_size, hidden_size1, hidden_size2, hidden_size3, output_size)

##################################
# Training a neural network model
# with RRELU activation function
##################################
nn_rrelu.train(X, y_one_hot, epochs=1001, lr=0.01)

Epoch 0: Loss 0.425362770010686, Accuracy 0.3496932515337423
Epoch 100: Loss 0.1722538021669186, Accuracy 0.901840490797546
Epoch 200: Loss 0.13053451403709015, Accuracy 0.9079754601226994
Epoch 300: Loss 0.11370320839132773, Accuracy 0.8957055214723927
Epoch 400: Loss 0.10982347564770722, Accuracy 0.901840490797546
Epoch 500: Loss 0.10483990809689325, Accuracy 0.9263803680981595
Epoch 600: Loss 0.10155752634211294, Accuracy 0.9325153374233128
Epoch 700: Loss 0.099518307514954, Accuracy 0.9202453987730062
Epoch 800: Loss 0.09799491669757482, Accuracy 0.9325153374233128
Epoch 900: Loss 0.09839336828285081, Accuracy 0.9263803680981595
Epoch 1000: Loss 0.09765980921693126, Accuracy 0.9263803680981595

##################################
# Plotting the computed gradients
# of the neural network model
# with RRELU activation function
##################################
plt.figure(figsize=(10, 6))
for key, value in nn_rrelu.gradients.items():
    plt.plot(value, label=key)
plt.title('RRELU Activation: Gradients by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Gradients')
plt.ylim(-0.05, 0.50)
plt.xlim(-50,1000)
plt.legend(loc="upper left")
plt.show()

##################################
# Plotting the computed cost
# of the neural network model
# with RRELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_rrelu.losses)
plt.title('RRELU Activation: Cost Function by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.ylim(0.05, 0.70)
plt.xlim(-50,1000)
plt.show()

##################################
# Plotting the computed accuracy
# of the neural network model
# with RRELU activation function
##################################
plt.figure(figsize=(10, 6))
plt.plot(nn_rrelu.accuracies)
plt.title('RRELU Activation: Classification by Iteration')
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.ylim(0.00, 1.00)
plt.xlim(-50,1000)
plt.show()

##################################
# Gathering the final values for 
# accuracy and loss error
# using a RRELU activation function
##################################
rrelu_metrics = pd.DataFrame(["ACCURACY","LOSS"])
rrelu_values = pd.DataFrame([nn_rrelu.accuracies[-1],nn_rrelu.losses[-1]])
rrelu_method = pd.DataFrame(["RRELU"]*2)
rrelu_summary = pd.concat([rrelu_metrics,
                           rrelu_values,
                           rrelu_method], axis=1)
rrelu_summary.columns = ['Metric', 'Value', 'Method']
rrelu_summary.reset_index(inplace=True, drop=True)
display(rrelu_summary)

	Metric	Value	Method
0	ACCURACY	0.92638	RRELU
1	LOSS	0.09766	RRELU

1.7. Consolidated Findings

1. This activation function showed the presence of the vanishing gradients problem resulting to a relatively higher loss and lower classification accuracy:
   • Sigmoid = Sigmoid Activation Function
2. These activation functions demonstrated the absence of the vanishing gradients problem resulting to a sufficiently low loss and higher classification accuracy:
   • RELU = Rectified Linear Unit Activation Function
   • Leaky_RELU = Leaky Rectified Linear Unit Activation Function
   • ELU = Exponential Linear Unit Activation Function
   • SELU = Scaled Exponential Linear Unit Activation Function
   • RRELU = Randomized Leaky Rectified Linear Unit Activation Function
3. The choice of Activation Function can significantly impact the performance and training dynamics of a neural network classification model, making it an important consideration during model development. Different activation functions have different properties, and selecting the appropriate one depends on factors such as the nature of the problem, the characteristics of the data, and the desired behavior of the network.

##################################
# Consolidating all the
# model performance metrics
##################################
model_performance_comparison = pd.concat([sigmoid_summary, 
                                          relu_summary,
                                          leakyrelu_summary, 
                                          elu_summary,
                                          selu_summary, 
                                          rrelu_summary], 
                                         ignore_index=True)
print('Neural Network Model Comparison: ')
display(model_performance_comparison)

Neural Network Model Comparison: 

	Metric	Value	Method
0	ACCURACY	0.748466	Sigmoid
1	LOSS	0.278053	Sigmoid
2	ACCURACY	0.932515	RELU
3	LOSS	0.096237	RELU
4	ACCURACY	0.926380	Leaky_RELU
5	LOSS	0.093777	Leaky_RELU
6	ACCURACY	0.926380	ELU
7	LOSS	0.094432	ELU
8	ACCURACY	0.901840	SELU
9	LOSS	0.096749	SELU
10	ACCURACY	0.926380	RRELU
11	LOSS	0.097660	RRELU

##################################
# Consolidating the values for the
# accuracy metrics
# for all models
##################################
model_performance_comparison_accuracy = model_performance_comparison[model_performance_comparison['Metric']=='ACCURACY']
model_performance_comparison_accuracy.reset_index(inplace=True, drop=True)
model_performance_comparison_accuracy

	Metric	Value	Method
0	ACCURACY	0.748466	Sigmoid
1	ACCURACY	0.932515	RELU
2	ACCURACY	0.926380	Leaky_RELU
3	ACCURACY	0.926380	ELU
4	ACCURACY	0.901840	SELU
5	ACCURACY	0.926380	RRELU

##################################
# Plotting the values for the
# accuracy metrics
# for all models
##################################
fig, ax = plt.subplots(figsize=(7, 7))
accuracy_hbar = ax.barh(model_performance_comparison_accuracy['Method'], model_performance_comparison_accuracy['Value'])
ax.set_xlabel("Accuracy")
ax.set_ylabel("Neural Network Classification Models")
ax.bar_label(accuracy_hbar, fmt='%.5f', padding=-50, color='white', fontweight='bold')
ax.set_xlim(0,1)
plt.show()

##################################
# Consolidating the values for the
# logarithmic loss error metrics
# for all models
##################################
model_performance_comparison_loss = model_performance_comparison[model_performance_comparison['Metric']=='LOSS']
model_performance_comparison_loss.reset_index(inplace=True, drop=True)
model_performance_comparison_loss

	Metric	Value	Method
0	LOSS	0.278053	Sigmoid
1	LOSS	0.096237	RELU
2	LOSS	0.093777	Leaky_RELU
3	LOSS	0.094432	ELU
4	LOSS	0.096749	SELU
5	LOSS	0.097660	RRELU

##################################
# Plotting the values for the
# loss error
# for all models
##################################
fig, ax = plt.subplots(figsize=(7, 7))
loss_hbar = ax.barh(model_performance_comparison_loss['Method'], model_performance_comparison_loss['Value'])
ax.set_xlabel("Loss Error")
ax.set_ylabel("Neural Network Classification Models")
ax.bar_label(loss_hbar, fmt='%.5f', padding=-50, color='white', fontweight='bold')
ax.set_xlim(0,0.40)
plt.show()

2. Summary

3. References

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• [Book] Deep Learning with Python by François Chollet
• [Book] The Elements of Statistical Learning by Trevor Hastie, Robert Tibshirani and Jerome Friedman
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• [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
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• [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
• [Book] Regression Modeling Strategies by Frank Harrell
• [Python Library API] NumPy by NumPy Team
• [Python Library API] pandas by Pandas Team
• [Python Library API] seaborn by Seaborn Team
• [Python Library API] matplotlib.pyplot by MatPlotLib Team
• [Python Library API] itertools by Python Team
• [Python Library API] operator by Python Team
• [Python Library API] sklearn.experimental by Scikit-Learn Team
• [Python Library API] sklearn.impute by Scikit-Learn Team
• [Python Library API] sklearn.linear_model by Scikit-Learn Team
• [Python Library API] sklearn.preprocessing by Scikit-Learn Team
• [Python Library API] scipy by SciPy Team
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• [Article] Missing Data Imputation Approaches | How to handle missing values in Python by Selva Prabhakaran (Machine Learning +)
• [Article] Master The Skills Of Missing Data Imputation Techniques In Python(2022) And Be Successful by Mrinal Walia (Analytics Vidhya)
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• [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
• [Article] Data Preprocessing in Python by Tarun Gupta (Towards Data Science)
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• [Article] Outlier Treatment with Python by Sangita Yemulwar (Analytics Vidhya)
• [Article] A Guide to Outlier Detection in Python by Sadrach Pierre (BuiltIn)
• [Article] How To Find Outliers in Data Using Python (and How To Handle Them) by Eric Kleppen (Career Foundry)
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• [Article] A Step by Step Backpropagation Example by Matt Mazur (MattMazur.Com)
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• [Article] Understanding Backpropagation in Neural Networks by Tech-AI-Math Team
• [Article] Backpropagation Neural Network using Python by Avinash Navlani (Machine Learning Geek)
• [Article] Back Propagation in Neural Network: Machine Learning Algorithm by Daniel Johnson (Guru99)
• [Article] What is Backpropagation? by Thomas Wood (DeepAI.Org)
• [Article] Activation Functions in Neural Networks [12 Types & Use Cases] by Pragati Baheti (V7.Com)
• [Article] Activation Functions in Neural Networks by Sagar Sharma (Towards Data Science)
• [Article] Comparison of Sigmoid, Tanh and ReLU Activation Functions by Sandeep Kumar (AItude.Com)
• [Article] How to Choose an Activation Function for Deep Learning by Jason Brownlee (Machine Learning Mastery)
• [Article] Choosing the Right Activation Function in Deep Learning: A Practical Overview and Comparison by Okan Yenigün (Medium)
• [Article] Activation Functions in Neural Networks by Geeks For Geeks Team
• [Article] A Practical Comparison of Activation Functions by Danny Denenberg (Medium)
• [Article] Activation Functions in Neural Networks: With 15 examples by Nikolaj Buhl (Encord.Com)
• [Article] Activation functions used in Neural Networks - Which is Better? by Anish Singh Walia (Medium)
• [Article] 6 Types of Activation Function in Neural Networks You Need to Know by Kechit Goyal (UpGrad.Com)
• [Article] Activation Functions in Neural Networks by SuperAnnotate Team
• [Article] Compare Activation Layers by MathWorks Team
• [Article] Activation Functions In Neural Networks by Kurtis Pykes (Comet.Com)
• [Article] ReLU vs. Sigmoid Function in Deep Neural Networks by Ayush Thakur (Wanb.AI)
• [Article] Using Activation Functions in Neural Networks by Jason Bronwlee (Machine Learning Mastery)
• [Article] Activation Function: Top 9 Most Popular Explained & When To Use Them by Neri Van Otten (SpotIntelligence.Com)
• [Article] 5 Deep Learning and Neural Network Activation Functions to Know by Artem Oppermann (BuiltIn.Com)
• [Article] Activation Functions in Deep Learning: Sigmoid, tanh, ReLU by Artem Oppermann
• [Article] 7 Types of Activation Functions in Neural Network by Dinesh Kumawat (AnalyticsSteps.Com)
• [Article] What is an Activation Function? A Complete Guide by Petru Potrimba (RoboFlow.Com)
• [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
• [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
• [Publication] Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification by Stef van Buuren (Statistical Methods in Medical Research)
• [Publication] Mathematical Contributions to the Theory of Evolution: Regression, Heredity and Panmixia by Karl Pearson (Royal Society)
• [Publication] A New Family of Power Transformations to Improve Normality or Symmetry by In-Kwon Yeo and Richard Johnson (Biometrika)
• [Course] IBM Data Analyst Professional Certificate by IBM Team (Coursera)
• [Course] IBM Data Science Professional Certificate by IBM Team (Coursera)
• [Course] IBM Machine Learning Professional Certificate by IBM Team (Coursera)
• [Course] Machine Learning Specialization Certificate by DeepLearning.AI Team (Coursera)

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